In traditional continuum mechanics, it can be shown using frame-invariance and the Cayley-Hamilton theorem that an isotropic, inelastic* fluid embedded in Galilean space-time must possess the following form of stress-strain rate relationship (if the relationship is analytic):
$$\bar{\bar{\sigma}} = \phi_0\bar{\bar{I}} + \phi_1\bar{\bar{D}} + \phi_2\bar{\bar{D}}\bar{\bar{D}}$$
where the strain rate $\bar{\bar{D}}$ is the symmetric part of the velocity gradient, $\bar{\bar{D}} = \frac{\nabla v + (\nabla v)^T}{2}$, $\bar{\bar{D}}\bar{\bar{D}} = D_{ij}D_{jk}$ in index notation, and the $\phi$'s are functions of the invariants of $\bar{\bar{D}}$. Fluids of these type are called Reiner-Rivlin fluids in the literature.
The Newtonian stress-strain rate model is obtained immediately by assuming $\phi_2 = 0$, and the rationale for doing so in the continuum mechanics literature is always that no fluid of this type as ever been experimentally observed with a nonzero $\phi_2$.
This leads me to believe that there is a physical restriction on $\phi_2$ that is perhaps outside the realm of continuum mechanics but not statistical mechanics. Is there any principle in statistical mechanics that forces a classical, isotropic, inelastic fluid to have a linear stress-strain rate relationship?
*Inelastic means the stress is only a function of the strain rate and not the strain itself.
